1. Projectile Motion Major Principles for All Circumstances Horizontal motion is constant velocity Vertical motion is constant downward acceleration a = g = -9.8 m/s 2

2. Projectile Motion The Big Four + One More In x x = v x t In y y = v oy t + 1/2 gt 2 v y = v oy + gt v y 2 = v oy 2 + 2gy y = 1/2(v oy + v y )t Ties them together Time & Launch Angle

3. Projectile Motion Types of Projectile Problems Type A -Half of a Parabola Type B - Type C - Full Parabola - Symmetric Partial or Asymmetric Parabola

4. Type A - Half of a Parabola Projectile Motion In the vertical direction The object acts like a dropped object Initial vertical velocity is zero; v oy = 0

5. Type A - Half of a Parabola Projectile Motion To solve for time, often you will use... y = v oy t + 1/2 gt 2 since v oy = 0 y = 1/2 gt 2 t =  (2y/g) Therefore... WARNING: Be careful using shortcut formulas!!!!

6. Type A - Half of a Parabola Projectile Motion If the problem is reversed... Romeo throws a rock up to Juliet; hits window horizontally Because of symmetry, just solve the problem backwards, make v oy = 0

7. Projectile Motion Type B - Full Parabola Notice the ball lands back in the truck... only if the truck moves with constant velocity

8. Projectile Motion Type B - Full Parabola If you solve for the full parabola... The vertical displacement is zero; y = 0 The time is the total hang time

9. Projectile Motion Type B - Full Parabola If you solve for half the parabola... The vertical velocity at the peak is; v y = 0 The time is equal to half the hang time

10. Projectile Motion Type B - Full Parabola The Range Formula vovo WARNING: Use the triangle for velocities only!!!! v oy = v o sin  v x = v o cos 

11. Projectile Motion Type B - Full Parabola The Range Formula v x = v o cos  x = v x t x = (v o cos  )t v oy = v o sin 

12. Projectile Motion Type B - Full Parabola The Range Formula x = v x t v oy = v o sin  x = (v o cos  )t v y = v oy + gt -v oy = v oy + gt -2v oy = gt -(2v oy )/g = t v y = -v oy -(2v o sin  )/g = t

13. Projectile Motion Type B - Full Parabola The Range Formula x = v x t x = (v o cos  )t -(2v o sin  )/g = t x = (v o cos  )(-2v o sin  /g)

14. Projectile Motion Type B - Full Parabola The Range Formula x = v x t x = (v o cos  )t x = (v o cos  )(-2v o sin  /g)

15. Projectile Motion Type B - Full Parabola The Range Formula x = v x t x = (v o cos  )t x = (v o cos  )(-2v o sin  /g) x = -v o 2 (2sin  cos  )/g Trig Identity: 2sin  cos  = sin2  x = -v o 2 sin2  /g

16. Projectile Motion Type B - Full Parabola The Range Formula x = v x t x = (v o cos  )t x = (v o cos  )(-2v o sin  /g) x = -v o 2 (2sin  cos  )/g x = -v o 2 sin2  /g

17. Projectile Motion Type B - Full Parabola The Range Formula x = v x t x = (v o cos  )t x = (v o cos  )(-2v o sin  /g) x = -v o 2 (2sin  cos  )/g x = -v o 2 sin2  /g WARNING: Be careful using shortcut formulas!!!!

18. Optimum Angle of 45  Maximum range Projectile Motion Type B - Full Parabola Supplementary Angles Equal ranges

19. Projectile Motion Type C - Partial or Asymmetric Parabola Each problem is unique, so take your time and... Some problems can be treated as two Type A problems stick to your major principles from the beginning

20. Projectile Motion Type C - Partial or Asymmetric Parabola We don’t know time, but we must find out the height (y) of an object. Very Unique Equation y = v oy t + 1/2gt 2

21. Projectile Motion Type C - Partial or Asymmetric Parabola Very Unique Equation y = v oy t + 1/2gt 2 vovo v oy = v o sin  v x = v o cos  y = (v o sin  )t + 1/2gt 2

22. Projectile Motion Type C - Partial or Asymmetric Parabola Very Unique Equation y = v oy t + 1/2gt 2 vovo v oy = v o sin  v x = v o cos  y = (v o sin  )t + 1/2gt 2 x = v x t x = (v o cos  )t x/(v o cos  ) = t

23. Projectile Motion Type C - Partial or Asymmetric Parabola Very Unique Equation y = v oy t + 1/2gt 2 y = (v o sin  )t + 1/2gt 2 x/(v o cos  ) = t y = v o sin  (x/(v o cos  )) + 1/2g(x/(v o cos  )) 2

24. Projectile Motion Type C - Partial or Asymmetric Parabola Very Unique Equation y = v oy t + 1/2gt 2 y = (v o sin  )t + 1/2gt 2 y = v o sin  (x/(v o cos  )) + 1/2g(x/(v o cos  )) 2 y = x(sin  /cos  ) + 1/2g(x 2 /(v o 2 cos 2  )) y = xtan  + gx 2 /(2v o 2 cos 2  )

25. Projectile Motion Type C - Partial or Asymmetric Parabola Very Unique Equation y = v oy t + 1/2gt 2 y = (v o sin  )t + 1/2gt 2 y = v o sin  (x/(v o cos  )) + 1/2g(x/(v o cos  )) 2 y = x(sin  /cos  ) + 1/2g(x 2 /(v o 2 cos 2  )) y = xtan  + gx 2 /(2v o 2 cos 2  ) Works for all types of problems, !! Most useful with Type C!!